Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,25)(24,27)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,28)(21,23)(22,27);(5,7,8,11)(6,10,9,12)(14,20)(15,24)(16,18)(21,22)(25,26)(27,28);(2,4)(13,16)(17,18)(19,27)(20,22)(23,28)(24,26);(6,9)(10,12)(13,16,17,18)(14,20,21,22)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^9|c^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72257307, #72263868, … |
| maximal normal subgroup of | #72239609, #72257307, #72263868, … |
| maximal quotient of | #72239716, #72264491 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422605);